Related papers: A positive characterization of rational maps

A branched covering $f: S^2 \to S^2$ is a nearly Euclidean Thurston (NET) map if each critical point is simple and its postcritical set has exactly four points. We show that up to equivalence, each NET map admits a normal form in terms of…

Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal…

We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and…

A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be a torus, a K3 surface, an Enriques surface or a non-minimal rational surface. We deal with results obtained in this last…

In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed…

One develops {\em ab initio} the theory of rational/birational maps over reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic. A numerical invariant of a rational map is introduced, called the Jacobian…

We explore distribution questions for rational maps on the projective line $\mathbb{P}^1$ over $\mathbb{Q}$ within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems…

Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP^2}$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an…

We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…

We show that if there exists a topologically expansive homeomorphism on a uniform space, then the space is always a regular space. Through examples we show that in general composition of topologically expansive homeomorphisms need not be…

Let $S$ be a surface with a metric $d$ satisfying an upper curvature bound in the sense of Alexandrov (i.e. via triangle comparison). We show that an almost conformal harmonic map from a surface into $(S,d)$ is a branched covering. As a…

In general a universal covering of a non connected topological group need not admit a topological group structure such that the covering map is a morphism of topological groups. This result is due to R.L. Taylor (1953). We generalise this…

We classify rational surfaces for which the image of the automorphisms group in the group of linear transformations of the Picard group is the largest possible. This answers a question raised by Arthur Coble in 1928, and can be rephrased in…

In this paper, we introduce cosine Thurston maps. In particular, we construct postsingularly finite topological cosine maps and focus on such maps with strictly preperiodic critical points. We use the techniques of Hubbard, Schleicher, and…

We describe an application of tropical moduli spaces to complex dynamics. A post-critically finite branched covering $\varphi$ of $S^2$ induces a pullback map on the Teichm\"uller space of complex structures of $S^2$; this descends to an…

We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on…

Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to…

We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by…

We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the relation of a hyperbolic 3-manifold to a…

In his 1979 paper Trotman proves, using the techniques of the Thom transversality theorem, that under some conditions on the dimensions of the manifolds under consideration, openness of the set of maps transverse to a stratification in the…